Detailed solutions please: 1. Let R and S be commutative rings and let sigma: R right arrow S be a homomorphism. Prove that if sigma is one to one and b in R is a zero divisor of R, then sigma(b) is a zero divisor of S. Solution we are given that R--->S is a homomorphism A homomorphism is a map that preserves selected structure between two algebraic structures, with the structure to be preserved being given by the naming of the homomorphism the set R and S are the same as they form a homomorphic relation. now b E R that is b lies with the set R and since R and S are homomorphic so b will also lie in the ser S since sigma is one to one function threfore the entire domain of sigma will be mapped . its also gven that sigma(b) is a zero divisor of R and since b E S as well so sigma(b) is a zero divisor of S as well hence proved..